Sparse Graphs for Belief Propagation Decoding of Polar Codes

Abstract

We describe a novel approach to interpret a polar code as a low-density parity-check (LDPC)-like code with an underlying sparse decoding graph. This sparse graph is based on the encoding factor graph of polar codes and is suitable for conventional belief propagation (BP) decoding. We discuss several pruning techniques based on the check node decoder (CND) and variable node decoder (VND) update equations, significantly reducing the size (i.e., decoding complexity) of the parity-check matrix. As a result, iterative polar decoding can then be conducted on a sparse graph, akin to the traditional well-established LDPC decoding, e.g., using a fully parallel sumproduct algorithm (SPA). This facilitates the systematic analysis and design of polar codes using the well-established tools known from analyzing LDPC codes. We show that the proposed iterative polar decoder has a negligible performance loss for shortto-intermediate codelengths compared to Arıkan's original BP decoder. Finally, the proposed decoder is shown to benefit from both reduced complexity and reduced memory requirements and, thus, is more suitable for hardware implementations.

Optimizing Polar Codes Compatible with Off-the-Shelf LDPC Decoders

“…As depicted in Fig. 4, both BER and BLER performance of the optimized codes under LDPC(-like) decoding achieve the performance of the 5G polar code under Arıkan's conventional BP decoder, with a performance gain of around 0.2 dB and 0.1 dB at BLER of 10 −2 and BER of 10 −3 , respectively, compared to the originally proposed code in [14]. They even outperform the 5G polar code under both Arıkan's conventional BP and plain SCL decoders towards a higher SNR range.…”

“…Polar codes can be interpreted as LDPC codes with an underlying sparse Tanner graph [14]. This is illustrated in Fig.…”

“…Assuming the schedule is now the same as the conventional flooding BP of LDPC codes, the graph resembles an LDPC code with some special constraint. In [14], it was given the name "LDPC-like" to address the fact that the channel information bits of this LDPC code are only at the last N VNs. Applying some basic pruning, the graph can be significantly reduced with an approximate reduction factor of 80%.…”

Polar Decoding on Sparse Graphs with Deep Learning

“…But the dense graph representation involved with many circles has demonstrated to show poor performance over additive white Gaussian noise (AWGN) channel [9]. Pruning methods for polar factor graph are consequently proposed in [9] to perform efficient polar decoding with LDPC-like manner. The sparse graph after using pruning techniques is shown as Fig.…”

“…In this work, we propose a sparse neural network decoder (SNND) for polar codes with high parallelism, low latency and low complexity. Inspired by [8], our SNND is constructed from the bipartite Tanner graph of polar codes in [9]. The sumproduct algorithm (SPA) is replaced by min-sum (MS) approximation to reduce complexity.…”